In the US, many PhD programs require applications to submit the results of one of the GRE subject tests. For example, almost every (serious) PhD program in mathematics that I've seen requires it. However, when looking at the admissions webpage of some programs, there sometimes seems to be some kind of grey area when it comes to the subject test.

The Department requires all applicants to submit GRE Mathematics Subject Test scores. Applicants should check on the ETS website for test dates in their area to insure the scores will be submitted before the application deadline.

So far so good, and if they left it at that, the message would seem to be: "you have to submit the GRE subject test, otherwise your application will be incomplete and rejected outright". However, it goes on to say:

While the admissions committee reviews all applications submitted by the deadline, missing math subject test scores are one less data point available to evaluate the application. Depending on the applicant pool and the strength of the application materials, the missing subject test scores may put the application at a disadvantage.

How should one interpret this? Does it mean:

"We thank you for the $105 donation, but your application lacks critical information and will almost certainly be rejected"; or

"Your application will be considered, provided it somehow demonstrates that you have a sufficiently strong mathematics background to succeed in our program"?

One one hand, it seems possible that they would want to "consider" applications without the GRE subject test, in that they receive the admission fee and that they can discard the application with very little time and effort (after all, they said the GRE was necessary). On the other hand, it seems silly that Harvard should reject an applicant who, by all accounts, seems to be this generation's Terence Tao, for the simple reason that he did not take the GRE subject test.

4 Answers
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This is certainly poorly and confusingly worded, but I think I can explain the reasoning based on similar issues I've seen with other departments. Here's my interpretation:

The department expects every applicant to take the GRE subject test. However, they know that every year a small fraction of the applicants screw up and don't manage to register in time to take it in the fall. It's impossible to take the test at the last minute, so at that point there's no way to fix the situation. The department doesn't want to automatically reject these applicants, for exactly the reason you mention (what if someone's the next Terry Tao?), and they are worried that if they list the GRE as an absolute requirement, then the university administration might force them to do that. On the other hand, they also don't intend to offer applicants the discretion to decide for themselves whether taking the subject test would strengthen their applications. They want everyone to take it, with the possibility of forgiveness for outstanding candidates who screw up. These goals get combined into the confusing statement that it's mandatory but your application can still be considered without it.

I don't know how the Harvard math department evaluates applicants with no GRE scores, but I can tell you how my department does. We're a little happier if we see an apologetic note about how the applicant intended to take the test but failed to do so in time, but of course we know such a note proves nothing. Our default assumption is that the applicant was worried about doing poorly, so we start asking questions like "How badly would we guess they might have done?" and "Would we have accepted them anyway, even with low GRE scores?" If the application is otherwise fantastic, then the chances of acceptance remain good. However, these are the cases in which we would have cheerfully made excuses for low scores (maybe the applicant just isn't good at standardized tests or had a bad day), so nothing is gained by skipping the exam. An otherwise marginal applicant with no GRE subject test score will be rejected, and a strong but not fantastic applicant will be at a disadvantage.

So my best guess as to what the policy means in practice is something like "if you don't take the exam, we'll be annoyed at your irresponsibility and we'll evaluate your application with a presumption that you wouldn't have done well on the exam, but we'll still consider admitting you." This means you have a strong incentive to take the exam if you can.

On one hand, it seems possible that they would want to "consider" applications without the GRE subject test, in that they receive the admission fee and that they can discard the application with very little time and effort

In a typical U.S. department, application requirements and graduate admissions decisions are handled pretty much entirely separately from application fees. For example, many people on the committee won't even know offhand what the current application fee is or where the money goes within the university. In particular, whoever wrote the policy about GRE scores probably wasn't thinking about application fees at all.

This is not a grey area. This is simply a contradiction. The first paragraph you've shared says it is required and the second sure seems to imply that they are only strongly recommended.

Since it's easier to use the single word "required" accidentally and sloppily than it is to write that second paragraph by accident, my guess would be that they mean that the subject GREs are only strongly recommended, not required.

Let's be very clear though: The opinion of people here doesn't matter. The specific departmental admissions officers and committee are the only opinions that matter since they will make the decision. If you see contradictory information on their website, point it out to them and ask for a clarification. They will clarify the issue for you and probably even fix the page.

Finally, even if something is "only" very strongly recommended, it's still strongly recommended! Top programs are extremely competitive and it's up to you to do everything you can to strengthen your application. You might be the next generation's superstar but unless you've already done incredible, it can be extremely hard to tell this. Fantastic scores on the subject GRE can help make your case. Not doing it at all might suggest some insecurity on your part that you might not do well. Take the test unless you have an extremely good reason not to.

There may be a conflict between the department and overall admissions requirements. For example in my Masters program the professors asked the admissions board to waive the GRE requirement for a strong candidate who applied too late to schedule the test (rolling admissions). However this was a unique situation and they would not have appealed the process unless the student in question was an extremely strong candidate (I believe that the GRE scores were out of date not missing, and that said student already had an MBA).
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kleinegJul 18 '14 at 17:50

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How is it a contradiction? "Required" does not specify the penalty. I've had classes where attendance was "required"; this didn't mean that you failed if you missed a class, merely that you lost some points. And I'm sure the police believe that obeying the speed limit is "required", but in practice, most people don't (in the U.S.).
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ruakhJul 19 '14 at 8:12

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@ruakh It seems to me that in the context of graduate school admissions (or similar competitions, like job applications), it is reasonable to assume that "condition X is required" should stand for "for your application to be considered, condition X must be met". I don't think that the penalty for not including other "required" items in a graduate application (transcript, letters of recommendation, statement of purpose, etc) is in any way ambiguous. As for the speed limit example, if you get caught disobeying the speed limit, you most definitely will suffer the penalty. Most people don't...
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user10761Jul 19 '14 at 15:43

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...obey the requirement because they know they won't get caught. If you disobey a requirement in a graduate admission, you will get caught.
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user10761Jul 19 '14 at 15:44

@RaisinBread: I don't think so. There are requirements, and then there are requirements. For all that we complain about bureaucracy and "process police", human systems generally allow some degree of flexibility and personal judgment. (And, no: trust me, speed limit laws are not strictly enforced in the States. It's not just a matter of catching people. Cops generally won't bother pulling someone over unless they're going at least 10 over -- sometimes more -- and even when they do pull someone over, they often issue a warning rather than a fine.)
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ruakhJul 20 '14 at 7:06

The answer by anonymous mathematician seems essentially correct to me.

One thing to remember is that (in all math depts. I'm aware of, including Harvard's, and mine) grad admissions are reviewed by a faculty committee, who have nothing to do with imposing/collecting application fees and so on --- so there is certainly no correlation between department rules/guidelines and things like fees.

The committee members will do their best to assess candidates based on all the information they have available, using their best judgement as to how well the candidates will perform in their program. At Harvard in particular, they have their pick of the very strongest candidates from all over the world. GRE's are just one small component of their evaluation process, but are useful --- if someone doesn't do very well on the GRE, this is a flag. Essentially all the students admitted by top programs such as Harvard will have gotten close to a perfect score on the GRE subject test, though, and so the GRE serves more as a negative indicator against admission (if a candidate didn't do we'll) than as a positive indicator that someone should be admitted.

But of course occasionally someone who is unquestionably very strong will have done poorly on the GRE, or not done it at all, and the department will want to have the option of being able to waive it as any kind of formal criterion. Hence the slightly ambiguous wording on their website.

(One thing to remember is the once the dept. committee makes its selection, there is a bureaucratic process within the university where the list of admitted candidates has to be given to/accepted by other university entities, such as the university's graduate school. The department probably has to give some explanation/justification for its selection, and some discussion of the criteria it used. The GRE is presumably one of these, but they don't want to put themselves in a straitjacket of not being able to ignore it in special cases, and hence --- as Anonymous Mathematician notes --- they have a slightly ambiguous, flexible, policy.)

"Essentially all the students admitted by top programs such as Harvard will have gotten close to a perfect score on the GRE subject test." I've known enough students with less than near-perfect scores who got into top programs to be pretty sure this isn't true. What I expect is true, though, is that essentially none of the students admitted by top programs have "poor" scores. So your point stands.
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Mark MeckesSep 10 '14 at 15:31

There is one more option that (I think) has not been mentioned. I know of schools where exam scores are absolutely required, but students are sometimes accepted even though scores were not yet submitted. At these schools, the student can be given a "conditional" acceptance, and has one semester to submit the required scores. If the student still doesn't have scores on file at the end of that first semester, the graduate school will not allow them to continue attending.

The same sort of conditional acceptance is used for students who need to submit final transcripts after they graduate from their undergraduate program. In most graduate programs, all students are required to submit final transcripts, but students can be accepted conditionally before the transcripts are received.

This kind of situation is suggested by language such as

While the admissions committee reviews all applications submitted by the deadline, missing math subject test scores are one less data point available to evaluate the application.

Note "submitted by the deadline". The school might decide that they will accept some students who don't submit GRE scores by the deadline, while still requiring that the students must eventually submit GRE scores. This allows the department to use the scores for statistical purposes even if the scores were not always used for admissions purposes.

This sort of conditional admission is related to a scheduling issue with the fall subject GRE. This year (2014), the last exam date is October 25, and scores are estimated to be mailed "approximately" on December 5. This is just barely enough time for students who apply to schools with a January 1 deadline and send the scores from the October exam immediately. If a student delays sending scores to a school, the scores might not arrive in time for a Jan 1 deadline. This is likely part of the meaning of the language

Applicants should check on the ETS website for test dates in their area to insure the scores will be submitted before the application deadline.

Students who send their scores a little late would probably be worse off if their application was summarily denied than if it is reviewed with missing GRE scores.